1. **Stating the problem:** We want to analyze the inequalities $2^n > n$ for $n \geq 1$ and $n \geq 2^n$ for all integers $n \geq 4$. 2. **Understanding the inequalities:** - The first inequality $2^n > n$ means the exponential function $2^n$ grows faster than the linear function $n$ starting from $n=1$. - The second inequality $n \geq 2^n$ for $n \geq 4$ suggests the linear function $n$ is greater than or equal to the exponential $2^n$ for integers $n$ starting at 4. 3. **Checking the first inequality $2^n > n$ for $n \geq 1$:** - At $n=1$, $2^1=2$ and $n=1$, so $2 > 1$ is true. - At $n=2$, $2^2=4$ and $n=2$, so $4 > 2$ is true. - For larger $n$, since $2^n$ grows exponentially and $n$ grows linearly, $2^n > n$ holds for all $n \geq 1$. 4. **Checking the second inequality $n \geq 2^n$ for $n \geq 4$:** - At $n=4$, $n=4$ and $2^4=16$, so $4 \geq 16$ is false. - At $n=5$, $5 \geq 32$ is false. - For larger $n$, $2^n$ grows faster than $n$, so $n \geq 2^n$ is false for all $n \geq 4$. 5. **Conclusion:** - The inequality $2^n > n$ is true for all integers $n \geq 1$. - The inequality $n \geq 2^n$ is false for all integers $n \geq 4$. Hence, the second inequality does not hold for $n \geq 4$. $$\boxed{\text{For } n \geq 1, \quad 2^n > n \text{ is true, but } n \geq 2^n \text{ is false for } n \geq 4.}$$